Tall sections from non-minimal transformations

In previous work, we have shown that elliptic fibrations with two sections,or Mordell-Weil rank one, can always be mapped birationally to a Weierstrassmodel of a certain form, namely, the Jacobian of a $\mathbb{P}^{112}$ model.Most constructions of elliptically fibered Calabi-Yau manifolds with twosections have been carried out assuming that the image of this birational mapwas a "minimal" Weierstrass model. In this paper, we show that for someelliptically fibered Calabi-Yau manifolds with Mordell-Weil rank-one, theJacobian of the $\mathbb{P}^{112}$ model is not minimal. Said another way,starting from a Calabi-Yau Weierstrass model, the total space must be blown up(thereby destroying the "Calabi-Yau" property) in order to embed the model into$\mathbb{P}^{112}$. In particular, we show that the elliptic fibrations studiedrecently by Klevers and Taylor fall into this class of models.